Foundation Pair Kernel Physical Source Covector Stationary At Coefficient Eq Phy

The source covector

A ledger, a discrete record of events, needs a rule for how a posting at one location changes the field at another. In Recognition Science, that rule is a linear functional: a machine that takes a field variation and returns a number, the source strength. The theorem stationaryAtCoefficient_eq_physicalSourceCovector proves that if such a rule is stationary at a dipole coefficient, then it must equal the explicit form sourceScale * (v a - v b). Here a and b are the two posting locations, v is the field variation, and sourceScale is a real number that sets the overall magnitude.

The result is a uniqueness theorem. It says that among all linear functionals satisfying the Gauss-source coupling law at a given scale, there is exactly one: the pair-kernel physical source covector. The library proves this in both directions. If a pairing is stationary at the dipole coefficient, it equals the covector; if it equals the covector, it is stationary. The covector itself is built from an integer Gauss source, then realified and coupled to field variations at the symbolic scale. The scale parameter is injective on a nontrivial posting, meaning different scales give different covectors, and the covector vanishes when the two posting locations coincide.

The theorem does not choose a numerical value for sourceScale. That freedom is deliberate: the Gauss law fixes the shape of the source as a dipole, but not its coupling magnitude. The module also does not derive OneLedgerLaw, and it does not identify the covector with a Noether momentum-map covector. That comparison is recorded as an ordinary equality proposition, MomentumMapEqualsPairKernelPhysicalSource, left for a later step. The proof is axiom-clean: no sorry, no new axiom, and no use of Constants.G, the Planck endpoint, or the relation source * hbar = 1.

/-- Stationarity at a coupling means work-response equals the constructed
covector.  The coupling remains an input; this does not select it.
`OneLedgerLaw` is the named special case at scale `1` and stays a named
premise, not a derived theorem. -/
theorem stationaryAtCoefficient_eq_physicalSourceCovector {n : ℕ}
    [DecidableEq (Fin n)]
    (w : Fin n → Fin n → ℝ) (e : Fin n → ℝ) (a b : Fin n)
    (sourceScale : ℝ)
    (hstat :
      PairKernelSourceCoupling.StationaryAtCoefficient w e
        (PairKernelSourceVariation.dipole a b) sourceScale) :
    ∀ v, PairKernelSourceCoupling.physSourcePairing w e v =
      pairKernelPhysicalSourceCovector sourceScale a b v := by
  intro v
  have hdrop :
      (∑ i, v i * PairKernelSourceVariation.dipole a b i) = v a - v b :=
    PairKernelSourceVariation.sum_mul_dipole v a b
  calc
    PairKernelSourceCoupling.physSourcePairing w e v =
        sourceScale *
          ∑ i, v i * PairKernelSourceVariation.dipole a b i :=
      hstat v
    _ = sourceScale * (v a - v b) := by rw [hdrop]
    _ = pairKernelPhysicalSourceCovector sourceScale a b v := rfl

/-- Uniqueness: any covector realizing the same Gauss/source-coupling law at
the same symbolic scale equals the constructed carrier. -/
theorem pairKernelPhysicalSourceCovector_unique {n : ℕ}
    (φ : PhysicalSourceCarrier n) (sourceScale : ℝ) (a b : Fin n)
    (hφ : RealizesPairKernelGaussSourceCoupling φ sourceScale a b) :
    φ = pairKernelPhysicalSourceCovector sourceScale a b := by
  apply LinearMap.ext
  intro v
  calc
    φ v =
        sourceScale *
          ∑ i, v i *
            PairKernelDiscreteGauss.divF
              (PairKernelDiscreteGauss.elementaryPosting a b) i :=
      hφ v
    _ = pairKernelPhysicalSourceCovector sourceScale a b v :=
          (pairKernelPhysicalSourceCovector_eq_divF_pairing
            sourceScale a b v).symm

/-- Distinct symbolic scales give distinct covectors on a nontrivial posting.
This is the precise sense in which scale freedom lives in the constructor:
Gauss fixes shape; scale remains a free real parameter. -/
theorem pairKernelPhysicalSourceCovector_scale_injective {n : ℕ}
    {a b : Fin n} (hab : a ≠ b) {s₁ s₂ : ℝ}
    (h : pairKernelPhysicalSourceCovector s₁ a b =
      pairKernelPhysicalSourceCovector s₂ a b) :
    s₁ = s₂ := by
  have hs :=
    congrArg
      (fun φ : PhysicalSourceCarrier n =>
        φ (fun i => if i = a then (1 : ℝ) else 0)) h
  have h1 := pairKernelPhysicalSourceCovector_source_basis s₁ a b hab
  have h2 := pairKernelPhysicalSourceCovector_source_basis s₂ a b hab
  simp only [h1, h2] at hs
  exact hs

What this page does not claim

The theorem does not select a numerical value for the source scale. The theorem does not derive OneLedgerLaw. The theorem does not identify the covector with a Noether momentum-map covector.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelPhysicalSourceCovector.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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